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19 <img class="icon32" alt="[\lambda\delta home]" title="\lambda\delta home" src="http://lambdadelta.info/images/crux_32.png" />
22 <div class="head1">cic:/matita/lambdadelta/basic_2/ (core λδ version 2)</div>
24 <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
26 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
29 <div xmlns:ld="http://lambdadelta.info/" class="text">
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33 <td class="snns capitalize italic sky">
34 <a href="http://lambdadelta.info/home.html">home</a>
36 <td class="snns capitalize italic magenta">
37 <a href="http://lambdadelta.info/news.html">news</a>
39 <td class="snns capitalize italic white">
40 <a href="http://lambdadelta.info/specification.html">specification</a>
42 <td class="snnn capitalize italic white">
45 <td class="snnn capitalize italic white">
48 <td class="snns capitalize italic orange">
49 <a href="http://lambdadelta.info/documentation.html">documentation</a>
51 <td class="snns capitalize italic green">
52 <a href="http://lambdadelta.info/implementation.html">implementation</a>
54 <td class="ssnn capitalize italic green">
59 <td class="snns capitalize sky">
60 <a href="http://lambdadelta.info/home.html#foreword">foreword</a>
62 <td class="snns capitalize magenta">
63 <a href="http://lambdadelta.info/news.html#milestones">milestones</a>
65 <td class="snns capitalize white">
66 <a href="http://lambdadelta.info/specification.html#v2">version 2</a>
68 <td class="snnn capitalize white">(<a href="http://lambdadelta.info/ground_2.html">background</a> - <a href="http://lambdadelta.info/basic_2.html">core</a> - <a href="http://lambdadelta.info/apps_2.html">applications</a>)</td>
69 <td class="snnn capitalize white">
72 <td class="snns capitalize orange">
73 <a href="http://lambdadelta.info/documentation.html#v2">version 2</a>
75 <td class="snns capitalize green">
76 <a href="http://lambdadelta.info/implementation.html#helena">helena</a>
78 <td class="ssnn capitalize green">
79 <a href="http://lambdadelta.info/osn/">Open Symbolic Notation (OSN)</a>
83 <td class="snss capitalize sky">
84 <a href="http://lambdadelta.info/home.html#citations">citations</a>
86 <td class="snss capitalize magenta">
87 <a href="http://lambdadelta.info/news.html#visibility">visibility</a>
89 <td class="snss capitalize white">
90 <a href="http://lambdadelta.info/specification.html#v1">version 1</a>
92 <td class="snsn capitalize white">(<a href="http://lambdadelta.info/ground_1.html">background</a> - <a href="http://lambdadelta.info/basic_1.html">core</a>)</td>
93 <td class="snsn capitalize white">(<a href="http://lambdadelta.info/static/matita/lambdadelta/">static HELM directory</a>)</td>
94 <td class="snss capitalize orange">
95 <a href="http://lambdadelta.info/documentation.html#v1">version 1</a>
97 <td class="snss capitalize green">
98 <a href="http://lambdadelta.info/implementation.html#lddl">library</a>
100 <td class="sssn capitalize green">(<a href="http://lambdadelta.info/static/lddl/">static LDDL directory</a>)</td>
106 <section>System's Syntax and Behavior</section>
107 <body>This is a summary of the "block structure"
108 of the System's syntactic items and reductions.
110 <table name="basic_2_blk"/>
111 <body>* In terms only.
112 ** In terms and local environments only.
113 *** In global environments only.
114 **** Sort level k in terms only.
117 <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="summary">Summary of the Specification <img class="icon37" alt="[butterfly]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
119 <div xmlns:ld="http://lambdadelta.info/" class="text">Here is a numerical account of the specification's contents
122 <div xmlns:ld="http://lambdadelta.info/" class="text">
123 <table cellpadding="4" cellspacing="0">
126 <td class="snns capitalize italic gray">category</td>
127 <td class="snns italic gray">objects</td>
128 <td class="snnn right italic gray">
131 <td class="snnn italic gray">
134 <td class="snnn right italic gray">
137 <td class="snnn italic gray">
140 <td class="ssnn right italic gray">
145 <td class="snns capitalize italic cyan">sizes</td>
146 <td class="snns italic cyan">files</td>
147 <td class="snnn right italic cyan">167</td>
148 <td class="snns italic cyan">characters</td>
149 <td class="snnn right italic cyan">167321</td>
150 <td class="snns italic cyan">nodes</td>
151 <td class="ssnn right italic cyan">875248</td>
154 <td class="snns capitalize italic green">propositions</td>
155 <td class="snns italic green">theorems</td>
156 <td class="snnn right italic green">47</td>
157 <td class="snns italic green">lemmas</td>
158 <td class="snnn right italic green">590</td>
159 <td class="snns italic green">total</td>
160 <td class="ssnn right italic green">637</td>
163 <td class="snss capitalize italic yellow">concepts</td>
164 <td class="snss italic yellow">declared</td>
165 <td class="snsn right italic yellow">23</td>
166 <td class="snss italic yellow">defined</td>
167 <td class="snsn right italic yellow">40</td>
168 <td class="snss italic yellow">total</td>
169 <td class="sssn right italic yellow">63</td>
174 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="B">Stage "B"</div>
175 <ul xmlns:ld="http://lambdadelta.info/">
177 <span class="emph alpha">Ongoing.</span>
178 Context-sensitive subject equivalence
179 for native type assignment.
182 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A2">Stage "A2": "Extending the Applicability Condition"</div>
183 <ul xmlns:ld="http://lambdadelta.info/">
185 <span class="emph alpha">2016 September 15.</span>
186 Confluence for context-sensitive parallel r-transition on terms.
189 <ul xmlns:ld="http://lambdadelta.info/">
191 <span class="emph alpha">2016 April 16.</span>
192 Grammatical component reconstructed:
193 grammar, relocation, s_transition, s_computation, static
194 (anniversary milestone).
197 <ul xmlns:ld="http://lambdadelta.info/">
199 <span class="emph alpha">2016 March 25.</span>
200 Relocation with reference transforming maps (rtmap).
203 <ul xmlns:ld="http://lambdadelta.info/">
205 <span class="emph alpha">2015 October 9.</span>
206 λδ version 2A2 is started.
209 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A1">Stage "A1": "Extending the Applicability Condition"</div>
210 <ul xmlns:ld="http://lambdadelta.info/">
212 <span class="emph delta">2015 August 27.</span>
213 λδ version 2A1 appears too complex and is dismissed.
216 <ul xmlns:ld="http://lambdadelta.info/">
218 <span class="emph gamma">2014 October 28.</span>
219 λδ version 2A1 is released.
222 <ul xmlns:ld="http://lambdadelta.info/">
224 <span class="emph beta">2014 September 9.</span>
225 Iterated static type assignment defined (more elegantly)
226 as a primitive notion.
229 <ul xmlns:ld="http://lambdadelta.info/">
231 <span class="emph beta">2014 June 18.</span>
232 Preservation of stratified native validity
233 for context-sensitive computation on terms.
236 <ul xmlns:ld="http://lambdadelta.info/">
238 <span class="emph alpha">2014 June 9.</span>
239 Strong qrst-normalization
240 for simply typed terms.
243 <ul xmlns:ld="http://lambdadelta.info/">
245 <span class="emph alpha">2014 April 16.</span>
246 Lazy equivalence on local environments
247 added as q-step to rst-computation on closures
248 (anniversary milestone).
251 <ul xmlns:ld="http://lambdadelta.info/">
253 <span class="emph alpha">2014 January 20.</span>
254 Parametrized slicing of local environments
255 comprises both versions of this operation
256 (one from basic_1, the other used in basic_2 till now).
259 <ul xmlns:ld="http://lambdadelta.info/">
261 <span class="emph alpha">2013 August 7.</span>
262 Passive support for global environments.
265 <ul xmlns:ld="http://lambdadelta.info/">
267 <span class="emph alpha">2013 July 27.</span>
268 Reaxiomatized β-reductum as in rt-reduction.
271 <ul xmlns:ld="http://lambdadelta.info/">
273 <span class="emph alpha">2013 July 20.</span>
274 Context-sensitive strong rt-normalization
275 for simply typed terms.
278 <ul xmlns:ld="http://lambdadelta.info/">
280 <span class="emph alpha">2013 April 16.</span>
281 Reaxiomatized substitution and reduction
282 commute with respect to subclosure
283 (anniversary milestone).
286 <ul xmlns:ld="http://lambdadelta.info/">
288 <span class="emph alpha">2013 March 16.</span>
289 Mutual recursive preservation of stratified native validity
290 for rst-computation on closures.
293 <ul xmlns:ld="http://lambdadelta.info/">
295 <span class="emph alpha">2012 October 16.</span>
296 Confluence for context-free parallel reduction on closures.
299 <ul xmlns:ld="http://lambdadelta.info/">
301 <span class="emph alpha">2012 July 26.</span>
302 Term binders polarized to control ζ-reduction (not released).
305 <ul xmlns:ld="http://lambdadelta.info/">
307 <span class="emph alpha">2012 April 16.</span>
308 Context-sensitive subject equivalence
309 for atomic arity assignment
310 (anniversary milestone).
313 <ul xmlns:ld="http://lambdadelta.info/">
315 <span class="emph alpha">2012 March 15.</span>
316 Context-sensitive strong normalization
317 for simply typed terms.
320 <ul xmlns:ld="http://lambdadelta.info/">
322 <span class="emph alpha">2012 January 27.</span>
323 Support for abstract candidates of reducibility.
326 <ul xmlns:ld="http://lambdadelta.info/">
328 <span class="emph alpha">2011 September 21.</span>
329 Confluence for context-sensitive parallel reduction on terms.
332 <ul xmlns:ld="http://lambdadelta.info/">
334 <span class="emph alpha">2011 September 6.</span>
335 Confluence for context-free parallel reduction on terms.
338 <ul xmlns:ld="http://lambdadelta.info/">
340 <span class="emph alpha">2011 April 17.</span>
341 λδ version 2 is started.
344 <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="structure">Logical Structure of the Specification <img class="icon37" alt="[butterfly]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
346 <div xmlns:ld="http://lambdadelta.info/" class="text">This table reports the specification's components and their planes.
348 <div xmlns:ld="http://lambdadelta.info/" class="text">
349 <table cellpadding="4" cellspacing="0">
352 <td class="snns top capitalize italic gray">component</td>
353 <td class="snns top italic gray">plane</td>
354 <td class="snns top gray">files</td>
355 <td class="snnn top gray">
358 <td class="snnn top gray">
361 <td class="ssnn top gray">
366 <td class="snns top capitalize italic water">rt-transition</td>
367 <td class="snns top italic water">t-bound context-sensitive rt-transition</td>
368 <td class="snns top water">lfpr ( ⦃?,?⦄ ⊢ ➡[?,?] ? )</td>
369 <td class="snnn top water">lfpr_length lfpr_drops lfpr_fqup lfpr_lfpx lfpr_lfpr</td>
370 <td class="snnn top water">
373 <td class="ssnn top water">
378 <td class="nnns top capitalize italic water">
381 <td class="nnns top italic water">
384 <td class="snns top water">cpr ( ⦃?,?⦄ ⊢ ? ➡[?] ? )</td>
385 <td class="snnn top water">cpr_drops</td>
386 <td class="snnn top water">
389 <td class="ssnn top water">
394 <td class="nnns top capitalize italic water">
397 <td class="nnns top italic water">
400 <td class="snns top water">cpm ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )</td>
401 <td class="snnn top water">cpm_simple cpm_drops cpm_lsubr cpm_cpx</td>
402 <td class="snnn top water">
405 <td class="ssnn top water">
410 <td class="nnns top capitalize italic water">
413 <td class="snns top italic water">uncounted context-sensitive rt-transition</td>
414 <td class="snns top water">lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )</td>
415 <td class="snnn top water">lfpx_length lfpx_drops lfpx_fqup</td>
416 <td class="snnn top water">
419 <td class="ssnn top water">
424 <td class="nnns top capitalize italic water">
427 <td class="nnns top italic water">
430 <td class="snns top water">cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )</td>
431 <td class="snnn top water">cpx_simple cpx_drops cpx_lsubr</td>
432 <td class="snnn top water">
435 <td class="ssnn top water">
440 <td class="nnns top capitalize italic water">
443 <td class="snns top italic water">counted context-sensitive rt-transition</td>
444 <td class="snns top water">cpg ( ⦃?,?⦄ ⊢ ? ⬈[?,?] ? )</td>
445 <td class="snnn top water">cpg_simple cpg_drops cpg_lsubr</td>
446 <td class="snnn top water">
449 <td class="ssnn top water">
454 <td class="snns top capitalize italic green">static typing</td>
455 <td class="snns top italic green">parameters</td>
456 <td class="snns top green">sh</td>
457 <td class="snnn top green">sd</td>
458 <td class="snnn top green">
461 <td class="ssnn top green">
466 <td class="nnns top capitalize italic green">
469 <td class="snns top italic green">restricted ref. for atomic arity assignment</td>
470 <td class="snns top green">lsuba ( ? ⊢ ? ⫃⁝ ? )</td>
471 <td class="snnn top green">lsuba_drops lsuba_lsubr lsuba_aaa lsuba_lsuba</td>
472 <td class="snnn top green">
475 <td class="ssnn top green">
480 <td class="nnns top capitalize italic green">
483 <td class="snns top italic green">atomic arity assignment</td>
484 <td class="snns top green">aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )</td>
485 <td class="snnn top green">aaa_drops aaa_fqus aaa_lfeq aaa_aaa</td>
486 <td class="snnn top green">
489 <td class="ssnn top green">
494 <td class="nnns top capitalize italic green">
497 <td class="snns top italic green">restricted ref. for local env.</td>
498 <td class="snns top green">lsubr ( ? ⫃ ? )</td>
499 <td class="snnn top green">lsubr_length lsubr_drops lsubr_lsubr</td>
500 <td class="snnn top green">
503 <td class="ssnn top green">
508 <td class="nnns top capitalize italic green">
511 <td class="snns top italic green">equivalence for closures on referred entries</td>
512 <td class="snns top green">ffeq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )</td>
513 <td class="snnn top green">ffeq_freq</td>
514 <td class="snnn top green">
517 <td class="ssnn top green">
522 <td class="nnns top capitalize italic green">
525 <td class="snns top italic green">equivalence for local environments on referred entries</td>
526 <td class="snns top green">lfeq ( ? ≡[?] ? )</td>
527 <td class="snnn top green">lfeq_length lfeq_lreq lfeq_fqup lfeq_lfeq</td>
528 <td class="snnn top green">
531 <td class="ssnn top green">
536 <td class="nnns top capitalize italic green">
539 <td class="snns top italic green">generic extension on referred entries</td>
540 <td class="snns top green">lfxs ( ? ⦻*[?,?] ? )</td>
541 <td class="snnn top green">lfxs_length lfxs_drops lfxs_fqup lfxs_lfxs</td>
542 <td class="snnn top green">
545 <td class="ssnn top green">
550 <td class="nnns top capitalize italic green">
553 <td class="snns top italic green">context-sensitive free variables</td>
554 <td class="snns top green">frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )</td>
555 <td class="snnn top green">frees_weight frees_lreq frees_drops frees_fqup frees_fqus frees_frees</td>
556 <td class="snnn top green">
559 <td class="ssnn top green">
564 <td class="snns top capitalize italic grass">s-computation</td>
565 <td class="snns top italic grass">iterated structural successor for closures</td>
566 <td class="snns top grass">fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )</td>
567 <td class="snnn top grass">fqus_weight fqus_drops fqus_fqup fqus_fqus</td>
568 <td class="snnn top grass">
571 <td class="ssnn top grass">
576 <td class="nnns top capitalize italic grass">
579 <td class="nnns top italic grass">
582 <td class="snns top grass">fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )</td>
583 <td class="snnn top grass">fqup_weight fqup_drops fqup_fqup</td>
584 <td class="snnn top grass">
587 <td class="ssnn top grass">
592 <td class="snns top capitalize italic yellow">s-transition</td>
593 <td class="snns top italic yellow">structural successor for closures</td>
594 <td class="snns top yellow">fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )</td>
595 <td class="snnn top yellow">fquq_length fquq_weight</td>
596 <td class="snnn top yellow">
599 <td class="ssnn top yellow">
604 <td class="nnns top capitalize italic yellow">
607 <td class="nnns top italic yellow">
610 <td class="snns top yellow">fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )</td>
611 <td class="snnn top yellow">fqu_length fqu_weight</td>
612 <td class="snnn top yellow">
615 <td class="ssnn top yellow">
620 <td class="snns top capitalize italic orange">relocation</td>
621 <td class="snns top italic orange">generic slicing for local environments</td>
622 <td class="snns top orange">drops_vector ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )</td>
623 <td class="snnn top orange">
626 <td class="snnn top orange">
629 <td class="ssnn top orange">
634 <td class="nnns top capitalize italic orange">
637 <td class="nnns top italic orange">
640 <td class="snns top orange">drops ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )</td>
641 <td class="snnn top orange">drops_lstar drops_weight drops_length drops_ceq drops_lexs drops_lreq drops_drops</td>
642 <td class="snnn top orange">
645 <td class="ssnn top orange">
650 <td class="nnns top capitalize italic orange">
653 <td class="snns top italic orange">generic relocation for terms</td>
654 <td class="snns top orange">lifts_vector ( ⬆*[?] ? ≡ ? )</td>
655 <td class="snnn top orange">lifts_lifts_vector</td>
656 <td class="snnn top orange">
659 <td class="ssnn top orange">
664 <td class="nnns top capitalize italic orange">
667 <td class="nnns top italic orange">
670 <td class="snns top orange">lifts ( ⬆*[?] ? ≡ ? )</td>
671 <td class="snnn top orange">lifts_simple lifts_weight lifts_lifts</td>
672 <td class="snnn top orange">
675 <td class="ssnn top orange">
680 <td class="nnns top capitalize italic orange">
683 <td class="snns top italic orange">ranged equivalence for local environments</td>
684 <td class="snns top orange">lreq ( ? ≡[?] ? )</td>
685 <td class="snnn top orange">lreq_length lreq_lreq</td>
686 <td class="snnn top orange">
689 <td class="ssnn top orange">
694 <td class="nnns top capitalize italic orange">
697 <td class="snns top italic orange">generic entrywise extension</td>
698 <td class="snns top orange">lexs ( ? ⦻*[?,?,?] ? )</td>
699 <td class="snnn top orange">lexs_length lexs_lexs</td>
700 <td class="snnn top orange">
703 <td class="ssnn top orange">
708 <td class="snns top capitalize italic red">grammar</td>
709 <td class="snns top italic red">append for local environments</td>
710 <td class="snns top red">append ( ? @@ ? )</td>
711 <td class="snnn top red">append_length</td>
712 <td class="snnn top red">
715 <td class="ssnn top red">
720 <td class="nnns top capitalize italic red">
723 <td class="snns top italic red">context-sensitive equivalences for terms</td>
724 <td class="snns top red">ceq</td>
725 <td class="snnn top red">ceq_ceq</td>
726 <td class="snnn top red">
729 <td class="ssnn top red">
734 <td class="nnns top capitalize italic red">
737 <td class="snns top italic red">same top term structure</td>
738 <td class="snns top red">tsts ( ? ≂ ? )</td>
739 <td class="snnn top red">tsts_tsts tsts_vector</td>
740 <td class="snnn top red">
743 <td class="ssnn top red">
748 <td class="nnns top capitalize italic red">
751 <td class="snns top italic red">closures</td>
752 <td class="snns top red">cl_weight ( ♯{?,?,?} )</td>
753 <td class="snnn top red">cl_restricted_weight ( ♯{?,?} )</td>
754 <td class="snnn top red">
757 <td class="ssnn top red">
762 <td class="nnns top capitalize italic red">
765 <td class="snns top italic red">internal syntax</td>
766 <td class="snns top red">genv</td>
767 <td class="snnn top red">
770 <td class="snnn top red">
773 <td class="ssnn top red">
778 <td class="nnns top capitalize italic red">
781 <td class="nnns top italic red">
784 <td class="snns top red">lenv</td>
785 <td class="snnn top red">lenv_weight ( ♯{?} )</td>
786 <td class="snnn top red">lenv_length ( |?| )</td>
787 <td class="ssnn top red">
792 <td class="nnns top capitalize italic red">
795 <td class="nnns top italic red">
798 <td class="snns top red">term</td>
799 <td class="snnn top red">term_weight ( ♯{?} )</td>
800 <td class="snnn top red">term_simple ( 𝐒⦃?⦄ )</td>
801 <td class="ssnn top red">term_vector ( Ⓐ?.? )</td>
804 <td class="nnns top capitalize italic red">
807 <td class="nnns top italic red">
810 <td class="snns top red">item</td>
811 <td class="snnn top red">
814 <td class="snnn top red">
817 <td class="ssnn top red">
822 <td class="nnss top capitalize italic red">
825 <td class="snss top italic red">external syntax</td>
826 <td class="snss top red">aarity</td>
827 <td class="snsn top red">
830 <td class="snsn top red">
833 <td class="sssn top red">
841 <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
843 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
846 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
847 <a href="http://validator.w3.org/check?uri=referer">
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